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## Category theory in Homological Algebra

### Tool and Object (2007-01-01) 32: 93-160 , January 01, 2007

Before around 1955, CT was almost exclusively used in algebraic topology and served there, at least up to Eilenberg and Steenrod, mainly as a conceptual (or linguistic) framework for the organization of a knowledge system. Arrows and arrow composition played an important role there, and the new framework emphasizing these aspects changed considerably the organization of topology as a whole (compare [Volkert 2002] chapter 6), but this change was rather a shift of emphasis from problem solving to conceptual clarification than direct progress in solving the problems formerly considered as central in the discipline (as, for instance, the classification of 3-manifolds). In the domain of algebraic topology, it was Kan who entered first a level of conceptual innovation on which CT came to serve also as a means of *deduction*. This means that results in the topological context have been obtained by the application of results established on the categorial level—results deeper than those available using solely the base concepts of category theory, *i.e*., results the proof (and already the formulation) of which used new, more involved concepts like adjoint functors and the general limit concept^{162}.

## Front Matter - Heavy-Tail Phenomena

### Heavy-Tail Phenomena (2007-01-01) , January 01, 2007

## Front Matter - High Performance Computing on Vector Systems 2006

### High Performance Computing on Vector Systems 2006 (2007-01-01) , January 01, 2007

## Riemannian geometry on the diffeomorphism group of the circle

### Arkiv för Matematik (2007-10-01) 45: 297-325 , October 01, 2007

The topological group
$\mathcal{D}^k(\mathbb{S})$
of diffeomorphisms of the unit circle
$\mathbb{S}$
of Sobolev class *H*^{k}, for *k* large enough, is a Banach manifold modeled on the Hilbert space
$H^k(\mathbb{S})$
. In this paper we show that the *H*^{1} right-invariant metric obtained by right-translation of the *H*^{1} inner product on
$T_{\rm id}\mathcal{D}^k(\mathbb{S})\simeq H^k(\mathbb{S})$
defines a smooth Riemannian metric on
$\mathcal{D}^k(\mathbb{S})$
, and we explicitly construct a compatible smooth affine connection. Once this framework has been established results from the general theory of affine connections on Banach manifolds can be applied to study the exponential map, geodesic flow, parallel translation, curvature etc. The diffeomorphism group of the circle provides the natural geometric setting for the Camassa–Holm equation – a nonlinear wave equation that has attracted much attention in recent years – and in this context it has been remarked in various papers how to construct a smooth Riemannian structure compatible with the *H*^{1} right-invariant metric. We give a self-contained presentation that can serve as a detailed mathematical foundation for the future study of geometric aspects of the Camassa–Holm equation.

## Optimization of interorbital three-dimensional transfer trajectories for stage spacecraft

### Automation and Remote Control (2007-08-01) 68: 1372-1390 , August 01, 2007

Problems of three-dimensional trajectory optimization of transfers for stage spacecraft and spacecraft with auxiliary fuel tank (AFT) from the low circuit orbit of the Earth’s artificial satellite (EAS) into the geostationary orbit and optimization problems of fuel distribution in stages or tanks are solved. Control of spacecraft motion is conducted by jet engines of bounded thrust; stage engines can have different characteristics, i.e., thrust-to-weight ratio and specific thrust. The used stage or auxiliary fuel tank is detached on the passive segment. Detachment is considered to be instantaneous, if the spacecraft position and velocity do not change at the detachment instant and the mass decreases in jumping mode. The mass of detached tanks is considered proportionate to the mass of consumed fuel; the mass of engine and auxiliary constructions, to thrust-to-weight ratio. The useful mass of the spacecraft with the limited time of transfer is maximized. The considered problems are intricate nonlinear optimal control problems with discontinuous phase variables. They are formalized as optimal control problems by a union of dynamic systems and are solved on the basis of the corresponding principle of the maximum. In this paper, boundary-value problems of the principle of the maximum are numerically solved by the shooting method. The choice of computing schemes of the shooting method and solution to systems of nonlinear equations is conducted by using a series of auxiliary problems.

## f-Vectors of Minkowski Additions of Convex Polytopes

### Discrete & Computational Geometry (2007-05-01) 37: 503-516 , May 01, 2007

The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes we call perfectly centered and the combinatorial properties of the Minkowski sum with their own dual. In particular, we have a characterization of the face lattice of the sum in terms of the face lattice of a given perfectly centered polytope. Exact face counting formulas are then obtained for perfectly centered simplices and hypercubes. The second type of results concerns tight upper bounds for the f-vectors of Minkowski sums of several polytopes.

## Odd Degree Polynomials on Real Banach Spaces

### Positivity (2007-02-01) 11: 143-153 , February 01, 2007

A classical result of Birch claims that for given *k, n* integers, *n*-odd there exists some *N* = *N(k, n*) such that for an arbitrary *n*-homogeneous polynomial *P* on
, there exists a linear subspace
of dimension at least *k*, where the restriction of *P* is identically zero (we say that *Y* is a null space for *P*). Given *n* > 1 odd, and arbitrary real separable Banach space *X* (or more generally a space with *w**-separable dual *X**), we construct an *n*-homogeneous polynomial *P* with the property that for every point 0 ≠ *x* ∈ *X* there exists some *k* ∈
such that every null space containing *x* has dimension at most *k*. In particular, *P* has no infinite dimensional null space. For a given *n* odd and a cardinal *τ* , we obtain a cardinal *N* = *N(τ, n*) = exp^{n}^{+1}*τ* such that every *n*-homogeneous polynomial on a real Banach space *X* of density *N* has a null space of density *τ* .

## Singularly perturbed system of differential equations with a rational singularity

### Differential Equations (2007-07-01) 43: 885-897 , July 01, 2007

## On the set of solutions to a variational phase transition problem of continuum mechanics

### Journal of Mathematical Sciences (2007-08-01) 144: 4645-4654 , August 01, 2007

A variational problem describing phase transitions is considered. It is shown that a multi-vaued function associating with the set of parameters the set of solutions to the problem is continuous and has compact values. Bibliography: 5 titles.

## The use of edge-directions and linear programming to enumerate vertices

### Journal of Combinatorial Optimization (2007-10-01) 14: 153-164 , October 01, 2007

Given a list of vectors that contains directions of the edges of a given polytope ℘ and the availability of an algorithm that solves linear programs over ℘, we describe a method for enumerating the vertices of ℘; in particular, the method is adaptable to polytopes which are presented as (linear) projections of polytopes having linear inequality representation. Polynomial complexity bounds under both the real and the binary computation models are derived when the dimension of the polytope is fixed and the given LP algorithm is polynomial.